Bounding the degree of solutions to Pfaff equations
type: Item
instance of: scholarly article
title: Bounding the degree of solutions to Pfaff equations
zbMATH Open document ID: 0986.32015
DOI: 10.5565/PUBLMAT_44200_10
author: Marco Brunella
author: Luís Gustavo Mendes
published in: Publicacions Matemàtiques
publication date: 2001-07-26T00:00:00Z
full work available at URL: https://eudml.org/doc/41409
review text: The authors investigate an old question asked by Poincaré: Given a foliation \(\mathcal F\) of degree \(d\) on an \(n\)-dimensional complex projective space and an \(\mathcal F\) -invariant projective hypersurface \(V\), is it possible to bound the degree of \(V\) by a number depending only on \(d\) and not on \(\mathcal F\) ? This question has been studied by various authors in the context of classical theory of foliations, e.g. by \textit{E. Ballico, M. Brunella, M. M. Carnicer, D. Cerveau} and \textit{A. Lins Neto}. In the present paper the authors work with Pfaff equations which are more general than foliations. Under a natural assumption that the ambient space \(X\) is a complex projective manifold with \(\text{Pic}(X)=\mathbb Z\) they give a positive answer to the above question in the case when \(V\) is a normal crossing hypersurface. At the end of the paper they discuss to which extent the normal crossing hypothesis can be weakened.
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 58A17
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 1563897
zbMATH Keywords: Pfaff equation
zbMATH Keywords: foliation
zbMATH Keywords: Poincaré problem
zbMATH Keywords: normal crossing hypersurface
reviewed by: Zbigniew Hajto
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1563897